The random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=\frac{1}{5}+p t+q t^{2}\)
where \(p\) and \(q\) are constants.
(a) Given that \(\mathrm{E}(X)=1.1\), find the numerical value of \(\operatorname{Var}(X)\).

The random variable \(Y\) has probability generating function \(\mathrm{G}_{Y}(t)\) given by
\(\mathrm{G}_{Y}(t)=\frac{2}{3} t\left(1+\frac{1}{2} t^{2}\right) .\)

The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
(b) Find the probability generating function of \(Z\).

(c) Find \(\mathrm{P}(Z\gt 2)\).

(d) State the most probable value of \(Z\).